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Failure of chaos control

van De Water W1, de Weger J

  • 1Physics Department, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|December 2, 2000
PubMed
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Controlling chaos in a pendulum experiment failed due to an unmeasured small eigenvalue. Experimental data distribution prevents accurate eigenvalue estimation, hindering chaos control efforts.

Area of Science:

  • Nonlinear dynamics
  • Experimental physics
  • Chaos theory

Background:

  • Parametrically excited pendulums can exhibit chaotic behavior.
  • Imperfections in excitation mechanisms introduce complexities, such as weakly excited degrees of freedom.
  • Controlling chaos is crucial for understanding and manipulating complex systems.

Purpose of the Study:

  • Investigate the challenges in controlling chaos in a parametrically excited pendulum with an imperfect excitation mechanism.
  • Identify the factors hindering successful chaos control in experimental settings.
  • Propose a condition for obtaining necessary information for chaos control.

Main Methods:

  • Experimental study of a parametrically excited pendulum with an imperfect excitation mechanism.

Related Experiment Videos

  • Numerical modeling to analyze system dynamics and eigenvalue estimation.
  • Time series analysis to examine phase-space distribution and waiting times.

    • Main Results:

    • Failure to control chaos despite a well-characterized pendulum state and weak perturbation.
    • Numerical models reveal the critical importance of a small eigenvalue, often overlooked.
    • Experimental time series data, distributed according to the natural measure, obscure the estimation of small eigenvalues.
    • Logarithmic oscillations observed in waiting times for control attempts due to phase-space distribution.

    Conclusions:

    • The presence of a small eigenvalue, arising from system imperfections, significantly impacts chaos control.
    • Accurate estimation of small eigenvalues from experimental time series is challenging due to data distribution.
    • Chaos must be suppressed before the requisite information for control can be reliably acquired.